Spectral projections enjoy high order convergence for globally smooth functions. However, a single discontinuity introduces O(1) spurious oscillations, Gibbs' Phenomena, and reduces the high order convergence rate to first order. We will show how adaptive mollifiers can be used to recover the high order convergence rate as well as remove the spurious oscillations found near a discontinuity. In addition, when these adaptive mollifiers are applied to an equidistant sampling of piecewise smooth functions we obtain an exponentially accurate "interpolation" scheme. This is a powerful new tool for equidistant data with applications in image processing and non-linear conservation laws. This research was conducted jointly with Eitan Tadmor.